Cross-spectral functions for the vertical and longitudinal components of turbulence of a two-dimensional gust field are derived from the point correlation function for turbulence due to von Karman. Closed form solutions in terms of Bessel functions of order 5/6 and 11/6 are found. An asymptotic expression for large values of the frequency argument, and series results for small values of frequency, are also given. These results now form the base for studying the effect of spanwise variations in turbulence for a turbulence environment which is characterized by the von Karman isotropic spectral relations. Previous studies were based mainly on the Dryden-type spectral representation

Houbolt, J. C.

Sen, A.

English

NASA Technical Reports Server

N A S A   C O N T R A C T O R  
R E P O R T  
CROSS-SPECTRAL FUNCTIONS 
BASED ON VON KARMAN'S 
SPECTRAL EQUATION 
by John C. Houbolt and Asim Sen 
Prepared by 
AERONAUTICAL RESEARCH ASSOCIATES OF PRINCETON, INC. 
Princeton, NJ. 08540 
f o r  Langley Research Center' 
N A T I O N A L   A E R O N A U T I C S   A N D   S P A C E   A D M I N I S T R A T I O N  W A S H I N G T O N ,  I?? C.'>$, e> .- ,MA.RtM'' 1972.' 
https://ntrs.nasa.gov/search.jsp?R=19720011379 2020-03-11T18:35:38+00:00Z
TECH LIBRARY KAFB, NM 
I 1. Report No. I 2. Government  Accession  No. I 3. Recipient's  Catalog  No. 
4. Title and Subtitle 5. Repor t  Date 
CROSS-SPECTRAL FUNCTIONS BASED ON VON KARMAN'S SPECTRAL EQUATION I March 1972 
6. Performing Organization Code 
7. Author(s) 8. Performing Organization Report No. 
John C .  Houbolt and Asim Sen 159 
10. Work Unit No. 
9. Performing Organization Name and Address 
Aeronau t i ca l   Research   Assoc ia tes   o f   P r ince ton ,   I nc .  
50 Washinqton Road 
11. Contract or Grant No. 
P r ince ton ,  NJ NAS 1-9200 
13. Type of Report and Period Covered 
12. Sponsoring Agency Name and Address 
Nat iona l  Aeronaut ics  and  Space A d m i n i s t r a t i o n  
Washington, DC 20546 
14. Sponsoring Agency Code 
15. Supplementary Notes 
16.  Abstract 
C r o s s - s p e c t r a l   f u n c t i o n s   f o r   t h e   v e r t i c a l  and l o n g i t u d i n a l  components o f  t u r b u l e n c e  o f  a 
t w o - d i m e n s i o n a l  g u s t  f i e l d  a r e  d e r i v e d  f r o m  t h e  p o i n t  c o r r e l a t i o n  f u n c t i o n  f o r  t u r b u l e n c e  
due t o  von Ka.rman. C l o s e d   f o r m   s o l u t i o n s   i n   t e r m s   o f   B e s s e l   f u n c t i o n s   o f   o r d e r  5 /6  and 11/6 
are  found. An asympto t i c   exp ress ion   f o r   l a rge   va lues  o f  the  frequency  argument,  and  series 
r e s u l t s   f o r   s m a l l   v a l u e s   o f   f r e q u e n c y ,   a r e   a l s o   g i v e n .  These r e s u l t s  now fo rm  the   base  fo r  
s t u d y i n g  t h e  e f f e c t  o f  s p a n w i s e  v a r i a t i o n s  i n  t u r b u l e n c e  f o r  a turbulence  env i ronment  
wh ich   i s   cha rac te r i zed   by   t he   von  Karman i s o t r o p i c   s p e c t r a l   r e l a t i o n s .   P r e v i o u s   s t u d i e s  
were  based  main ly   on  the  Dryden- type  spect ra l   representat ion.  
". " ~ ~ 
17. Key Words (Suggested by Author(s)) 
Cross-Spectra l   Funct ions,  Von Karman's  Spectral  
18. Distribution Statement 
U n c l a s s i f i e d  -- U n l i m i t e d  Equat ions,   Atmospher ic   Turbulence  Representat ion 
20. Security Classif. (of this page) 22. Price' 21. No. of Pages 
U n c l a s s i f i e d  U n c l a s s i f i e d  17 $3.00 
For sale by the  National Technical Information Service, Springfield, Virginia 22151 
p' 
I 
CROSS-SPECTRAL  FUNCTIONS  BASED ON 
V O N - K ~ R M ~ N  1s SPECTRAL EQUATION 
by John C.  Houbolt and ' A s i m  Sen 
Aeronautical  Research Associates of Pr inceton,  Inc.  
ABSTRACT 
C r o s s - s p e c t r a l  f u n c t i o n s  f o r  t h e  v e r t i c a l  and l o n g i t u d i n a l  
components of turbulence of a two-dimensional  gust  f ie ld  are  
der ived  f rom the  poin t  cor re la t ion  func t ion  for  tu rbulence  due t o  
von K&m&. Closed form solut ions in  terms of Besse l  func t ions  
of  order 5/6 and 11/6 are  found. An asymptot ic  expression for  
large values of the frequency agreement,  and s e r i e s  r e s u l t s  f o r  
small values   of   f requency,   are   a lso  given.   These  resul ts  now 
form the  base  for  s tudying  the  e f fec t  o f  spanwise  var ia t ions  in  
turbulence  for  a turbulence environment which i s  charac te r ized  by  
the  von K&m&n i s o t r o p i c   s p e c t r a l   r e l a t i o n s .   P r e v i o u s   s t u d i e s  
were based mainly on the Dryden-type spectral  representat ion.  
INTRODUCTION 
In  r e fe rences  1 and 2 de r iva t ions  a re  g iven  fo r  t he  c ros s -  
s p e c t r a  f o r  a two-dimensional gust  f ield based upon the  poin t  
c o r r e l a t i o n  and s p e c t r a l  f u n c t i o n s  which  were o f t en  used  in  gus t  
s t u d i e s  a t  t h a t  t i m e ;  s p e c i f i c a l l y ,  the po in t  func t ions  due t o  
Dryden  and  which y i e l d  a 2 behavior a t  high  frequency for the  
spectrum  were  employed. More r ecen t   gus t   s tud ie s   i nd ica t e  that 
a c o r r e l a t i o n  and s p e c t r a l  f u n c t i o n  due t o  von K&m& - one  which 
i n d i c a t e s  a behavior a t  h igh  f requencies  - seems t o  be a 
b e t t e r  model for   s tabi l ized  a tmospheric   turbulence.   Study of t h i s  
func t ion  revea led  that  i t  too could be extended to the two- 
d imens iona l  gus t  f ie ld  case  in  c losed  form fash ion .  The purpose 
of t h i s  r e p o r t  i s  t o  p r e s e n t  t h e  d e r i v a t i o n  and the c r o s s - s p e c t r a l  
f u n c t i o n s  t h a t  a p p l y  f o r  t h i s  case.  
0 
& 
I .. . .  . 
SYMBOLS 
r 
t 
U 
v 
X 
Z 
v 
0 
modif ied Bessel  funct ion of  the f i r s t  kind 
modified Bessel function of the second kind 
i n t e g r a l  s c a l e  of turbulence 
c o r r e l a t i o n  d i s t a n c e  
p o i n t  c o r r e l a t i o n  f u n c t i o n  
c r o s s - c o r r e l a t i o n   f u n c t i o n   f o r  w component 
c r o s s - c o r r e l a t i o n   f u n c t i o n   f o r  u component 
separat ion dis tance between paths  1 and 2 
v e l o c i t y  
c o r r e l a t i o n   d i s t a n c e  V t  
genera l   var iab le  
reduced  frequency - uiil v 
nondimensional  separat ion dis tance 
rms value  of w turbulence component 
S 
power s p e c t r a  
c ros s - spec t r a  
angular  f r e que ne y 
2 
CROSS-SPECTRA FOR TWO-DIMENSIONAL  GUST  STRUCTURE  BASED  ON 
THE VON KARMAN SPECTRAL FUNCTION 
A s  i n  r e f e r e n c e  1, c o n s i d e r  t h e  g u s t  f i e l d  t o  b e  homogenous 
and  i so t rop ic  and momentarily  frozen  (Taylor  hypothesis).  As such, 
the  cor re la t ion  func t ion  should  be  independent  of any f l i g h t  tra- 
verse  path taken.  With re ference  to  the  fo l lowing  ske tch ,  
1 2 
co r re l a t ion  be tween  gus t  ve loc i t i e s  a long  pa ths  1 and 2 should 
thus be associated w i t h  t he  co r re l a t ion  d i s t ance  r =/s2+ V 2 t 2  
i n   p l a c e  of t h e   c o r r e l a t i o n   d i s t a n c e  x of t he   po in t   co r re l a t ion  
func t ion .  
The von K&m& c o r r e l a t i o n  f u n c t i o n  f o r  v e r t i c a l  g u s t  ve- 
l o c i t i e s  i s  g i v e n  by 
where u = , and  where L i s  t h e   i n t e g r a l   s c a l e   l e n g t h .  To 
de r ive  the  c ros s - spec t r a  be tween  the  ve r t i ca l  ve loc i t i e s  a long  
pa th  1 and t h e  v e r t i c a l  v e l o c i t i e s  a l o n g  p a t h  2,we rep lace  x by 
r and  hence  def ine  the  cross-correlat ion  funct ion as 
X 
1 3391; 
where u = - 1 339 
(5 
S ( J = -  L 
By means of 
For l a t e r  u s e ,  t h e  f a c t  i s  a l s o  n o t e d  t h a t  
The c r o s s - s p e c t r a l  f u n c t i o n  f o l l o w s  from equation (4 )  as 
-00 
For tuna te ly ,  th i s  equat ion,   wi th R represented  by equat ion (4) 
has  a closed form integrat ion ( t ran&?orm 941 i n  r e f e r e n c e  3 ) .  
Af te r  some manipulation and use of equat ion  (5), t h e  r e s u l t  f o r  
%2 for  the  case  under  cons idera t ion  i s  found t o  be 
4 
2 = + (1.339v). where 1.339 
( J = -  S L 
Because 
equat ion (7)  may a l s o  be w r i t t e n  i n  t e r m s  of the frequency argu- 
ment v as 
This  i s  t h e  b a s i c  r e s u l t  of the  present  paper .  I t s  eva lua t ion  i s  
encumbered because of the  uncommon f r a c t i o n a l  o r d e r s  for the  
Bessel   funct ions.  This eva lua t ion  and assoc ia ted   curves   a re  d i s -  
c u s s e d  i n  a subsequent  sect ion.  
A s  a check, equation (8) should reduce t o  the one-dimensional 
s p e c t r a l   r e s u l t  when 0 = 0 . This reduct ion  i s  as follows. For 
z small 
K n ( z )  = 2 ( n  - l)! z ; n f o  n-1  -n 
With t h i s  r e s u l t ,  e q u a t i o n  (8) becomes 
5 
L 
1 
From t he  known r e l a t i o n s  
n! = r ( n  + 1) = n r ( n )  
= 1.339 
th i s  equat ion  reduces ,  as i t  should, t o  the  fo l lowing  wel l  known 
r e s u l t  f o r  the one-dimensional case 
6 
EVALUATION O F  EQUATION (8) 
Since no tables were  found for  the  Besse l  func t ions  of order  
6 and 
cont inuous property of Besse1 , func t ions  wi th  respec t  t o  t h e i r  
o rder .  The  known closed  form  half-order   funct ions were introduced 
as follows: 
5 11 , evalua t ion  was made by cons ide r ing   t he   ana ly t i ca l ly  
- e z  ~ 5 / 2  ( z )  = 1 + 2 + -  ~ 5 / 2 ( ~ )  - 2 
3 3  
Z 
Similar  expressions were w r i t t e n  fo r  t he  func t ions  of order  1 and 
2. Then f o r  a given  value of z , a p l o t  of y ( z )  versus n 
was made, and f rom the  resu l t ing  curve  values o? yn(z )  a t  
n = 5 and  n = were  read  by  interpolation. The values of 
the Bessel  funct ion needed in  equat ion (8)  then followed from the 
r e l a t i o n s  
11 
Spec i f i c  numer i ca l  r e su l t s  found  fo r  t he  c ros s  spec t r a  a re  
shown i n  f i g u r e  1. These may b e  u s e d  d i r e c t l y  i n  s t u d i e s  where 
va r i a t ion  in  gus t  ve loc i t i e s  ac ross  the  span  i s  be ing  t r ea t ed .  
The fn te rpola t ion  technique  as used here  for  evaluat ion was aided 
by   cons ider ing   the   asymptot ic   express ion   for   equa t ion  (8) and 
t h e  r e s u l t s  a t  very  low z , as d i scussed  in  the  nex t  s ec t ion .  
Asymptotic - .~ r e s u l t  -~ and expres s ion   fo r  low z .- For l a r g e  z 
7 
8 
-1 -r t 
I I -  
t 1- 
I 
Y 
OL 
v -
I .  
Figure 1. Cross-Spectra for Treatment of Nonuniform 
Spanwise Gusts 
With this r e su l t ,  equa t ion  (8) reduces to  the fol lowing asymptot ic  
r e s u l t ,   a p p l i c a b l e  a t  l a r g e  z 
)8’3[8 (1.339) 2 0 5/3 z -413 
3 
Figure 2 gives  a comparison of results as obtained from t h i s  
equat ion and  from  equation (8) .  It i s  seen that the  asymptotic 
r e s u l t   s e r v e s   q u i t e   w e l l   f o r  v > 1 . 
A f a c t  of i n t e re s t ing  phys ica l  s ign i f i cance  can  be brought out 
by consider ing the behavior  of t he  c ros s - spec t r a l  func t ion  a t  l a r g e  
frequency  argument i n  a par t icular   nondimensional   manner .   In  
response  s tudies  i t  i s  of ten convenient  or i n s t r u c t i v e  to express  
r e s u l t s  i n  t e r m s  of t h e  r a t i o  of t he  c ros s  spec t r a  to t he  po in t  
spec t ra ,  that  i s  , equat ion (8) divided by equation (10) . With the  
use of equat ion (11) ins t ead  of equat ion ( 8 ) ,  i t  can  be shown that 
for high frequencies  th i s  r a t i o  i s  simply 
It i s  noted that  the r a t i o  i s  a func t ion  of the frequency argument 
V 
i t  can be reasoned that ce r t a in  r e sponse  phenomena may be expected 
t o  peak at a c e r t a i n  - Us r ega rd le s s  of the  sca le  of turbulence.  
For  example, i n  t h e  c a s e  o f  r o l l i n g  r e s p o n s e  due. to spanwise 
v a r i a t i o n s  i n  v e r t i c a l  g u s t  v e l o c i t i e s ,  maximum r o l l i n g  power may 
be expected, as e x p r e s s e d  i n  t h e  s p e c t r a l  r a t i o  f o r m ,  when 
i s  near T , where s i s  taken as the  wing  span. 
O S  - only;  the  scale L no longer  appears.  Because of this f a c t ,  
V 
- U S  
V 
A use fu l   exp res s ion   fo r  small z i s  a l so   r ead i ly   found  
through use of t he  fo l lowing  de f in i t i ons .  
9 
Through means of  these express ions ,  equat ion  (8)  may be  expanded i n t o  
a s e r i e s   i n v o l v i n g  0 and z , with  t h e   f o l l o w i n g   r e s u l t .  
1 1 / 3  + .18(;) (-1 -t - 10
- ,9770 513 (1  + - 3 z  + w z  +68816 22  z6 + ...) 2 9 4  
- . o y j 6 0 ~ ~ / ? ( 1  + -& z2 + q m Z  9 4  +i€mTzT z6 + ...) 
(14) 
This  equat ion  was used  to  check  the  eva lua t ion  of the  c ross -  
s p e c t r a l   f u n c t i o n s  a t  low values  of z . 
10 
2 
Q W  
Figure 2a. Cross-Spectra f o r  0 = 0.2 . 
11 
1.0 
IO" 
I o - ~  
IO+ 
I 
12 
0-2 IO" 1.0 IO IO2 
W L  
Figure  2b. Cross-Spectra for cr = 0.4 . 
I .o 
IO" 
I o - ~  
I d 4  
lo-* 10- I 
i I 
I I 
= 1. 
I 
I .o 
OL 
V 
=- 
Figure 2 c .  Cross-Spectra for CJ = 0.6 . 
14 
OL 
V 
y =- 
I .o IO 
Figure 2 d .  Cross-Spectra f o r  CJ = 0.8 . 
1.0 
I o-2 
I o - ~  
-~ I i I i i i i  i i i~ i 1 i i 1 1 i  I i t l i  i ~ i i i i i  I i i i  I 1 1 1 1 c t  
3- 1.0 IO IO 
OL 
V 
y =- 
Figure 2 e .  Cross-Spectra f o r  (5 = 1.0 . 
CROSS  PECTRA  FOR LONGITUDINAL TURBULENCE 
The p o i n t  c o r r e l a t i o n  f u n c t i o n  f o r  t h e  l o n g i t u d i n a l  component 
of tu rbulence   tha t   cor responds  t o  equat ion  (1) i s  given  by 
U (15) 
X where u = 1.339L . I f  x i s  rep laced  by ' r =d- as 
done wi th  equat ion  ( 2 ) ,  t he  fo l lowing  c ross -co r re l a t ion  func t ion  
a p p l i c a b l e  t o  l o n g i t u d i n a l  t u r b u l e n c e  i s  obta ined .  
where u = ____ 0 /- and 0 = - S 1.339 L O  From t h i s  func t ion  
t h e  c r o s s - s p e c t r a l  f u n c t i o n  for l ong i tud ina l  t u rbu lence  i s  found 
t o  be 
CONCLUDING REMARKS 
This  paper has p resen ted  c losed  fo rm so lu t ions  fo r  t he  c ros s -  
spec t ra l   func t ions   based  on von K&rm&nls spec t r a l  equa t ion .  These 
r e l a t i o n s  and assoc ia ted  curves  thus  form a new and more appropr ia te  
b a s e  f o r  s t u d y i n g  t h e  e f f e c t s  of spanwise  var ia t ions  in  turbulence  
on a i r c r a f t  g u s t  r e s p o n s e .  
16 
REFERENCES 
1. Houbolt,  John C. "On t4fe Response of Structures Having 
Multiple Random Inputs. Jahr. 1957 der WGL, Fr iedr .  Vieweg 
& Sohn (Braunschweig) , pp 296-305. 
2. Diederich,  Franklin W. and Drischler, Joseph A. "Effect of 
Spanwise Variations i n  Gust In tens i ty  on the L i f t  Due to 
Atmospheric  Turbulence. NACA TN 3920, A p r i l  1957. 


Cross-spectral functions based on von Karman's spectral equation

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